Migration based on the one-way acoustic wave equation has the subject of several proposals, after the former was successfully introduced in the early 1970's [Claerbout, J., "Numerical Holography", Acoustical Holography, Vol. 3, A. F. Metherell, Ed., 273-283, Plenum Press, New York, 1970; and "Toward a Unified Theory of Reflector Mapping", Geophysics, 36, 467-481, 1971]. The most promising of these proposals included the following:
(i) F-K Migration Techniques
[Stolt, R. H., "Migration by Fourier Transform", Geophysics, 43, 23-48, 1978];
(ii) Kirchhoff Migration Systems
[Schneider, W. S., "Integral Equation Formulation For Migration in Two and Three Dimensions", Geophysics, 43, 49-76, 1978]; and
(iii) Space-derivative Migration Techniques
[Gazdag, J., "Wave Equation Migration With the Accurate Space Derivative Method", Geophysical Prospecting, 28, 60-70, 1980].
Although each of the proposals, supra, has strengths and weaknesses (depending at least in part on the exploration situation and processing purpose), the methods in toto have at least two drawbacks, viz., (i) each is based on the theory of acoustic wave propagation, i.e. transference of P-waves only; and (ii) there is an assumption that the data had been stacked before migration occured, that is to say, it is assumed the reflector was not steep and the media velocity had been laterally constant. These assumptions are necessary to make data "source-receiver coincident" data so that the latter can be migrated without use of any particular imaging principles.
While the disadvantages, listed as items (i) and (ii) supra, can be overcome by using a migration before stacking technique (with imaging being based on the reciprocity theorem, after Schultz, P. S. and Sherwood, J. W. C., "Depth Migration Before Stack", Geophysics, 45, 376-393, 1980), the first-mentioned disadvantage has remained and has become even more of a problem with the advent of collection of seismic data using multicomponent receivers since the collected data contains both P- and S-wave data subject to conversion within the subsurface under survey.
While it has been recognized that elastic wave (rather than acoustic) migration offers the advantage of providing final results having higher signal-to-noise ratio characteristics (since a larger portion of wave energy would be migrated) none of the proposals of which I am aware have been successfully implemented using stable processing algorithms, have solved P- and S-wave couplings and conversions of dipping reflectors or utilized a wave tracking concept based on finite time differences between adjacent estimations.
In this regard I am aware of the following additional research that bears on the method of the present invention:
(i) a finite element migration proposal for multicomponent data that does not include wave tracking; and does not solve the problem posed by P- and S-wave conversions at dipping reflectors [Marfurt, K. J., "Elastic Wave Equation Migration-inversion", Ph.D. Thesis, Columbia University, 1978];
(ii) a Kirchhoff migration proposal for flat reflectors that does not include wave tracking; nor solve the problem of P- and S-wave conversion at dipping reflectors [Wang, M. Y. and Kuo, J. T., "Implementation of the Simple P and S Simultaneous Migration Method", Project MIDAS Annual Report II, 1-35, Columbia University, 1981]; and
(iii) a one-way elastic wave inversion proposal for refraction and reflection data that, although providing velocity and scattering matrix information, does not include wave tracking; solve the problem of multicomponent receivers with P- and S-waves at dipping reflectors; nor compensate for lateral velocity changes. [Clayton, R. W., "Wavefield Inversion Methods for Refraction and Reflection Data", Ph.D. Thesis, Stanford University, 1981].